Mathematical Physics

I will explain how to obtain the Gordon-Stafford construction and some related constructions of Z-algebras in the literature, using certain mathematical avatars of line defects in 3d N=4 theories. ​Time permitting, I will discuss the K-theoretic and elliptic cases as well.

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory--Lurie proposed a conjectural substitute, later termed the fundamental local equivalence, relating categories of arc-integrable Kac--Moody representations and Whittaker D-modules on the affine Grassmannian. With a few exceptions, we verified this conjecture non-factorizably, as well as its extension to the affine flag variety. This is a report on joint work with Justin Campbell and Sam Raskin.

This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate's ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures: our results amount to an equivalence of A/B-twists of the free hypermultiplet and a U(1)-gauged hypermultiplet.

We will review a set of conjectures related to the structure of cohomological Hall algebras (COHA) of categories of Higgs sheaves on curves. We then focus on the case of P^1, and relate its COHA to the affine Yangian of sl_2.

We describe an algorithm which takes as input any pair of permutations and gives as output two permutations lying in the same Kazhdan-Lusztig right cell. There is an isomorphism between the Richardson varieties corresponding to the two pairs of permutations which preserves the singularity type. This fact has applications in the study of W-graphs for symmetric groups, as well as in finding examples of reducible associated varieties of sln-highest weight modules, and comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. This is joint work with Peter McNamara.

A fundamental theorem in the theory of Vertex algebras (known as Zhu’s theorem) demonstrates that the space generated by the characters of certain Vertex algebras is a representation of the modular group. We will cast this theorem in the language of homotopy theory using the language of conformal blocks. The goal of this talk is to justify the claim that equivariant elliptic cohomology, seen as a derived spectrum, is a homotopical analog of Zhu’s theorem in the special case of the Affine Vacuum vertex algebra at a fixed integral level. The talk will not require knowing the definition of Vertex algebras or conformal blocks.

Quasi-elliptic cohomology is closely related to Tate K-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories. It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal. In addition, we define twisted quasi-elliptic cohomology. They can be related to a twisted equivariant version Devoto’s elliptic cohomology via a Chern character map. Moreover, we construct twisted Real quasi-elliptic cohomology and the Chern character map in this case. This is joint work with Matthew Spong and Matthew Young.

In this talk I will discuss an interesting phenomenon, namely a correspondence between sigma models and vertex operator algebras, with the two related by their symmetry properties and by a reflection procedure, mapping the right-movers of the sigma model at a special point in the moduli space to left-movers. We will discuss the examples of N=(4,4) sigma models on $T^4$ and on $K3$. The talk will be based on joint work with Vassilis Anagiannis, John Duncan and Roberto Volpato.

We study elliptic characteristic classes of natural subvarieties in some ambient spaces, namely in homogeneous spaces and in Nakajima quiver varieties. The elliptic versions of such characteristic classes display an unexpected symmetry: after switching the equivariant and the Kahler parameters, the classes of varieties in one ambient space ``coincide” with the classes of varieties in another ambient space. This duality gets explained as “3d mirror duality” if we regard our ambient spaces as special cases of Cherkis bow varieties. I will report on a work in progress with Y. Shou, based on earlier related works with G. Felder, A. Smirnov, V. Tarasov, A. Varchenko, A. Weber, Z. Zhou.

(Super)conformal algebras on two-dimensional spacetimes play a ubiquitous role in representation theory and conformal field theory. In most cases, however, superconformal algebras are finite dimensional. In this talk, we introduce refinements of certain deformations of superconformal algebras which share many facets with the ordinary (super) Virasoro algebras. Representations of these refinements include the higher dimensional Kac—Moody algebras, and many more motivated by physics. Finally, we will show how these algebras enhance to higher dimensional chiral/factorization algebras which upon ``localization" deform to well-studied chiral algebras on Riemann surfaces.